Pardon implicit-atlas specialization to Floer cylinder moduli

For a closed symplectic manifold $(M,\omega )$ and $n\ge 0$, put ${\mathrm{\Delta }}^n=\{(x_1,\dots ,x_n):0\le x_1\le \cdots \le x_n\le 1\}$, with ${\mathrm{\Delta }}^0$ a point. A Floer parameter simplex is a smooth family $H_{\sigma }:{\mathrm{\Delta }}^n\times M\times S^1\to \mathbb{R}$ of one-periodic Hamiltonians and a smooth family $J_{\sigma }:{\mathrm{\Delta }}^n\to \mathcal{J}(M)$ of $\omega$-tame almost-complex structures, both constant near the vertices. Every vertex Hamiltonian has a supplied smooth time-one evolution and all its one-periodic orbits are nondegenerate; every vertex $J$ is compatible. Use the rational capped generators and reference weights $a(p)=-{\mathcal{A}}_H(p)$ of def-gts-rational-capped-hamiltonian-floer-module. Under its local cap convention, recapping by $A$ increases the energy weight by $\omega (A)$ and changes degree by $-2c_1(A)$. Fix the dimension convention below as part of the geometric package, including its Fredholm index and determinant-line identifications.

Define $f_n(x)={\sum }_{j=1}^n\cos (\pi x_j)$ and the vector field $V_n=-\mathrm{\nabla }{f}_n={\sum }_{j=1}^n\pi \sin (\pi x_j){\mathrm{\partial }}_{x_j}$, using the Euclidean metric on the displayed coordinates. It is tangent to every face, has the vertices as its critical points, and is directed from vertex $0$ (index $n$) to vertex $n$ (index $0$). For $n\ge 1$, let $F({\mathrm{\Delta }}^n)$ be the space of reduced broken $V_n$-flow lines from vertex $0$ to vertex $n$, modulo independent translation of each nonconstant segment. A representative is a finite list with strictly increasing endpoint vertices $0=v_0<\cdots <v_k=n$; it has no constant segment. For $n=0$ define $F({\mathrm{\Delta }}^0)$ to be a singleton. Equivalently, if constants are allowed in a parameter-list representative, identify lists exactly by insertion/deletion of constant segments and translation of the remaining segments. This equivalence concerns the parameter space only.

Here are explicit cube coordinates and their topology. Put $\varphi (s)=\frac{2}{\pi }\arctan (e^{{\pi }^{2}s})$. An unbroken flow line has $x_j(s)=\varphi (s+a_j)$ with $a_1\le \cdots \le a_n$, unique up to a common additive constant. Its $i$th coordinate, indexed by the intermediate vertex $i$, is $b_i=1-\exp (-(a_{n-i+1}-a_{n-i}))$. Permit $b_i=1$ to mean a break at vertex $i$, and use the same finite differences on each unbroken block. Thus every reduced broken list corresponds uniquely to $[0,1{]}^{n-1}$; $n=1$ gives a point. Give $F$ this cube topology. A break product inserts $b_i=1$, and a face omitting an intermediate vertex imposes the corresponding equality of adjacent $x_j$, namely $b_i=0$; the remaining ordered coordinates are inherited. These are the compatible product/face maps.

A stable Floer trajectory of type $(\sigma ,p,q)$ consists of a connected nodal genus-zero complex curve $C$ with two ordered punctures $x^{-}$ and $x^{+}$, a finite ordered parameter list $\mathrm{\ell }=({\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_k)$, and a map building $u$. The list has weakly increasing endpoint vertices $0=v_0\le \cdots \le v_k=n$, permits constant segments at equal vertices, and its reduction is a point $\bar{\mathrm{\ell }}\in F({\mathrm{\Delta }}^n)$. Constants remain recorded when they accompany cylindrical levels of the building; deleting a constant parameter segment alone does not identify two map buildings. For $n=0$ every parameter segment is constant at the sole vertex. Normalize the domain at every node, including the nodes at which sphere trees leave the unique dual-graph path from $x^{-}$ to $x^{+}$. The normalized components on that path determine $k$ ordered cylindrical levels. Each is a smooth genus-zero component with two distinguished path ends (the original punctures or branches of adjacent path nodes); deleting just those two points identifies it with $\mathbb{R}\times S^1$. Branches of off-path bubble nodes remain ordinary marked attachment points on this cylinder. The other normalized components form the attached sphere trees; they are not included as subsets of a cylindrical level. The map on level $a$ is a smooth $u_a:\mathbb{R}\times S^1\to M$ satisfying ${\mathrm{\partial }}_s{u}_a+J_{\sigma ({\mathrm{\ell }}_a(s))}(u_a)({\mathrm{\partial }}_t{u}_a-X_{H_{\sigma ({\mathrm{\ell }}_a(s)),t}}(u_a))=0.$ Its finite energy is $E_F(u_a)={\int }_{\mathbb{R}\times S^1}\omega ({\mathrm{\partial }}_s{u}_a,{\mathrm{\partial }}_t{u}_a-X_{H_{\sigma ({\mathrm{\ell }}_a(s)),t}}(u_a))dsdt<\mathrm{\infty }$; it is nonnegative because the equation gives ${\mathrm{\partial }}_t{u}_a-X_H=J_{\sigma ({\mathrm{\ell }}_a(s))}{\mathrm{\partial }}_s{u}_a$ and each $J_{\sigma }$ is $\omega$-tame. The negative end of level $a$ is asymptotic to a capped orbit ${\gamma }_{a-1}$ of the vertex Hamiltonian at $v_{a-1}$ and its positive end to ${\gamma }_a$ at $v_a$, with ${\gamma }_0=p$ and ${\gamma }_k=q$. Adjacent levels use the same intermediate orbit and cap. A sphere bubble attached at a point where ${\mathrm{\ell }}_a(s_i)\to x\in {\mathrm{\Delta }}^n$ is a smooth sphere map $v$ satisfying $dv\circ j=J_{\sigma (x)}(v)\circ dv$; its node value equals the value of the level or parent sphere. The capping class on each level, together with its attached sphere classes, is the class specified by its incoming and outgoing capped generators. Each cylindrical level carries its displayed angular coordinate $t\in S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ as part of the data. An isomorphism on level $a$ is exactly $(s,t)\mapsto (s+c_a,t)$ for a real constant $c_a$, with $u_a(s,t)=u_a^{\mathrm{\prime }}(s+c_a,t)$ and ${\mathrm{\ell }}_a(s)={\mathrm{\ell }}_a^{\mathrm{\prime }}(s+c_a)$. On the attached spheres it is a biholomorphism preserving the sphere maps and attachment points; it also preserves the two ends, nodes, level order, orbit labels and caps. Stability means that the resulting self-isomorphism group is finite. In particular rotations of $t$ are not isomorphisms, even when the Hamiltonian and trajectory are independent of $t$. This is the graph-map convention of Pardon Definition 10.2.2, where the isomorphism acts identically on the target $M\times S^1$ and only translates its real level factors. The reduced cap group on a component $M_{\alpha }$ is ${\pi }_2(M_{\alpha })\mathrm{/}(\ker {\omega }_{\alpha }\cap \ker c_1)$, equivalently the image of $({\omega }_{\alpha },2c_1)$.

The Gromov topology on these trajectory classes compares the parameter paths and maps jointly. Its convergence criterion is the following: (i) the reduced parameter lines $\bar{\mathrm{\ell }}$ converge in the compact cube $F({\mathrm{\Delta }}^n)$, with the prescribed vertex breaks and matching discrete asymptotic orbit and cap labels; (ii) for each limiting cylindrical level $a$, there are compact-core embeddings into the approximating cylinders of the form ${\iota }_{N,a}(s,t)=(s+c_{N,a},t)$, on domains exhausting that level away from nodes, such that both $u_N\circ {\iota }_{N,a}\to u_a$ and ${\mathrm{\ell }}_N(s+c_{N,a})\to {\mathrm{\ell }}_a(s)$ in $C^{\mathrm{\infty }}$ on compact subsets. The same constants $c_{N,a}$ must be used for both comparisons, including when the Hamiltonian and almost-complex families are constant on the simplex. If several limiting levels come from one approximating cylinder, their core embeddings are ordered and their relative translations diverge in that order. End tails converge to the specified periodic orbits and parameter vertices. (iii) At every concentration point, nested holomorphic coordinate embeddings exhaust the plane of each attached sphere away from its node points, with the rescaled maps converging in $C^{\mathrm{\infty }}$ on compact subsets, recursively for every descendant sphere, as in the fixed-source rescaling convention above. The parameter value at each attachment is the limit of the parameter path in the same cylinder coordinates; descendants inherit that value. (iv) All node values match and no energy is lost on cylinder-breaking or sphere-node necks. Precisely, the total geometric energy converges to the sum of the energies of the limiting cylindrical levels and the symplectic areas of all limiting sphere components; the energy on the complements of increasing compact cores tends to the corresponding limiting end tails. The topology on the quotient is generated by these joint compact-core $C^r$ comparisons for all integers $r\ge 0$, the cube-coordinate neighborhoods for $F({\mathrm{\Delta }}^n)$, the discrete orbit/cap conditions and the energy-tail bounds. Only simultaneous translations of a parameter path, its map and attachment coordinates are allowed in these comparisons; the reduced parameter cube alone does not record their relative position. Domain breaking is represented by neck lengths tending to infinity, and sphere bubbling by the nested rescalings above. In this topology a Pardon Floer implicit-atlas package is required to have compact Hausdorff stable-trajectory spaces; compactness and Hausdorffness are package properties, not consequences merely of naming the topology.

A Pardon Floer implicit-atlas package consists of compact Hausdorff stable-trajectory spaces $\mathcal{M}(\sigma ,p,q)$ with the Gromov topology, their breaking and simplex-face stratifications and compatible product/face maps, and an equivariant system of finite-dimensional reductions. A thickening datum has a compact codimension-compatible transverse submanifold $D_{\alpha }\subset M\times S^1\times \sigma$, a finite permutation group on its marked intersections, a finite-dimensional real obstruction space, and an equivariant perturbation section supported away from nodes and asymptotic markings. Thickened trajectories record the transverse intersections and solve the perturbed equation. The compatible charts obey the footprint, overlap, covering and topological submersion axioms; their virtual orientation lines have coherent trivializations under the product/face maps. For a boundary stratum $\eta$ the prescribed virtual dimension is $\mathrm{vdim}\mathcal{M}(\sigma ,p,q{)}_{\le \eta }=\mathrm{\mid }q\mathrm{\mid }-\mathrm{\mid }p\mathrm{\mid }+\dim \sigma -1-\mathrm{codim}(\eta )\mathrm{.}$ All sphere strata must be included in this package and its exact virtual boundary identities. Expected complex smoothing codimension alone is not a replacement for those identities.

The filtered finiteness requirements are that for every fixed $\sigma ,p,R$ there are only finitely many capped $q$ with $\mathcal{M}(\sigma ,p,q)\mathrm{\ne }\mathrm{\varnothing }$ and $a(q)<a(p)+R$, and for every $\sigma$ there is a finite $L_{\sigma }$ such that $a(q)-a(p)\ge L_{\sigma }$ whenever this moduli space is nonempty. Nonzero virtual coefficients must be supported on nonempty spaces. These are geometric requirements in addition to Novikov ring completeness.

A coherent virtual chain realization, which is additional data required to construct operators from this atlas, consists of rational homological chain complexes $W_{\ast }(\sigma ,p,q)$ with differential $\mathrm{\partial }$ of degree $-1$, compatible degree-zero evaluation chain maps $ϵ:W_{\ast }\to \mathbb{Q}[0]$, product/face chain maps, and chains ${\lambda }_{\sigma ,p,q}\in W_{\mathrm{\mid }q\mathrm{\mid }-\mathrm{\mid }p\mathrm{\mid }+n-1}(\sigma ,p,q),n=\dim \sigma \mathrm{.}$ Set $ϵ$ to zero on degrees other than zero. Evaluation commutes with the face maps and multiplies on products in their specified order. With those maps understood, the exact signed boundary identity is $\mathrm{\partial }{\lambda }_{\sigma ,p,r}={\sum }_{k=0}^n{\sum }_q(-1{)}^{k+\mathrm{\mid }p\mathrm{\mid }}{\lambda }_{\sigma \mathrm{\mid }[0,\dots ,k],p,q}\cdot {\lambda }_{\sigma \mathrm{\mid }[k,\dots ,n],q,r}-{\sum }_{k=1}^{n-1}(-1{)}^{k+\mathrm{\mid }p\mathrm{\mid }}{\lambda }_{\sigma \mathrm{\mid }[0,\dots ,\widehat{k},\dots ,n],p,r}\mathrm{.}$ The products on the right are pushed into $W_{\ast }(\sigma ,p,r)$, and the sums there are finite; operator coefficient sums satisfy the filtered finiteness requirements above. All choices are equivariant under simultaneous recapping of their endpoints. For this definition, the evaluated operator means $f_{\sigma }(p)={\sum }_qϵ({\lambda }_{\sigma ,p,q})q$, with only degree-zero chains contributing; write $d_i=f_i$ on a vertex. The chain realization also supplies the following normalization: for every degenerate parameter edge at vertex $i$, its evaluated operator differs from the identity by $d_i{B}_i+B_i{d}_i$ for a continuous finite matrix $B_i$ of degree $-1$ over the graded rational completion, where $d_i$ is the evaluated vertex operator. This normalization is a supplied geometric/virtual-chain property, not a formal consequence of the displayed boundary identity. Chains on triangles comparing forward, reverse and degenerate edges must be coherently chosen on their shared edges.

Pardon's Theorem 10.3.5 establishes the named equivariant atlas in the source's closed Hamiltonian Floer setting; Proposition 10.3.3 gives local regularity and coherent orientations, and Lemma 10.3.4 gives domain stabilization. His §§7.5–7.7 construct coherent virtual choices and the resulting rational complex up to chain homotopy, including the degenerate-edge normalization. Convention 7.0.1 requires all atlas group orders to be invertible in the ground ring; rational coefficients satisfy this requirement. The source grading character is $\mathrm{g}\mathrm{r}(h)=2⟨c_1,h⟩$ on its deck class $h$ (Definition 10.2.3). The local capped-generator convention above assigns $-2c_1(A)$ to the class $A$ defined by $w\mathrm{\#}A$; this definition does not identify $h$ with $A$. A concrete transfer of the source atlas to these local generators must state the deck-group identification and verify its energy and grading characters. No such sign identification is inferred from the shared notation. The source uses almost-complex structures independent of $t$. One may also specify $t$-dependent compatible vertex structures and tame simplex families, but this is an additional conditional regime requiring the same complete package; its construction is not attributed here to Theorem 10.3.5. This definition supplies neither a local reconstruction of the analytic/virtual construction nor a differential on the separate ${\mathbb{F}}_2$ module.

Rational capped Hamiltonian Floer modules and their graded submodules

Stable almost-complex maps and Gromov convergence

Zinger fixed-domain interior Gromov compactness

Pardon §10.2.3(iv) states compactness and the required finiteness by appeal to Gromov compactness. The local fixed-source closed-domain theorem is contextual; it proves neither breaking on noncompact cylinders nor the family compactification. A local proof must still supply asymptotic convergence, cylinder breaking, bubble extraction, neck energy and class conservation, compact Hausdorff quotient topology, and the filtered finiteness bounds for the parameter families.

Appendix C proves the topological gluing and orientation comparisons, using periodic-orbit orientation results of Floer–Hofer. Specifically C.13.2 invokes orientation triviality from the remark after their Theorem 2 and associative gluing from Theorem 10; the existing source record also identifies Proposition 9. The current local source record reports metadata/paywall access rather than full text of Floer–Hofer. No new full-text access or substitute proof is claimed here. The Fredholm determinant-line triviality, recapping equivariance, obstruction-space compatibility and boundary-sign comparison remain local construction obligations, as do the weighted gluing estimates and virtual-chain evaluation/product construction. The general source theorem is established literature; these are missing local proofs, not newly asserted open problems.